It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $$A_\infty $$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $$ \Omega \subset \mathbb {R}^{n+1}$$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $$\Omega $$ , with data in $$L^p(\partial \Omega )$$ for some $$p<\infty $$ . In this paper, we give a geometric characterization of the weak- $$A_\infty $$ property, of harmonic measure, and hence of solvability of the $$L^p$$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.

中文翻译：

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谐波测度和定量连通性：狄利克雷问题的 $$L^p$$-可解性的几何特征

众所周知，在开集 $$ \Omega \subset 的边界上，调和测度相对于表面测度的定量的、尺度不变的绝对连续性（更准确地说，是弱 $$A_\infty $$ 属性） \mathbb {R}^{n+1}$$ 带有Ahlfors-David 正则边界，等价于$$\Omega $$ 中狄利克雷问题的可解性，数据在$$L^p(\partial \Omega )$$ 一些 $$p<\infty $$ 。在本文中，我们给出了弱 $$A_\infty $$ 属性、调和测度的几何特征，从而给出了对某些有限 p 的 $$L^p$$ Dirichlet 问题的可解性。这种特征是在自然的背景假设（内部开瓶器条件，以及边界的 Ahlfors-David 规律性）下获得的，并且在某种意义上是最佳的：我们提供了其中一个都不存在的反例（或者甚至是两者之一，上界或下界，Ahlfors-David 边界）；此外，这些例子表明，上下 Ahlfors-David 界限在数量上都是尖锐的。